If and are locally free, then need not be. This is a fundamental example.

If and are locally free, then is too. This is important and not too hard.

If and are locally free *and finite rank*, then is too. This is also useful.

But if and are locally free, *and infinite rank*, then need not be locally free. The purpose of this post is to give **Daniel Litt‘**s explanation to me of a simple example of this. I thought this example was not right to include in the Rising Sea (it would be distracting), but I wanted to preserve it for posterity (and for myself).

I will describe a surjective map of free modules whose kernel is not locally free. Viewed geometrically, this provides a surjective map of trivial (infinite-dimensional) vector bundles whose kernel is not a vector bundle. Of course, the kernel is projective.

Let be a projective module over a ring which is not locally free. (Such things exist, see for example the Stacks Project. To be explicit, there and is the ideal (where there are 1’s).

Then is a direct summand of a free module, say . Writing

gives as a direct summand of a free module with free complement. Then there is a short exact sequence

exhibiting as the kernel of a surjective map of free modules, where the map is the projection.

]]>As always, I’m happy to hear of any errors/typos/suggestions!

]]>If you are interested in learning about associated points, or solidifying your understanding, or getting a more geometric view of them, please take a look — it is ten pages, and intended to be quite readable to those who have read the first five chapters or so of The Rising Sea.

The new exposition attempts to more directly follow the geometric point of view, set out well for example by Matthew Emerton (perhaps on this blog — if I can find the link I will add it here). I was trying to do something with my previous exposition, but did not succeed, and I think this works better. But I am very interested in hearing what people think, who are reading it right now.

This should be something that you can work through in an evening, and discuss with someone else. I am hoping there will be a couple of epiphanies in there. I am certain there will be typos!

The point of view is summarized here:

(For some reason the letter “p” was deleted in the last line of the pdf above…)

I guess there is no harm in putting the entire thing here, in case someone feels like skimming through it on this page.

]]>I want to share a proof of the Cohomology and Base Change theorem that really makes it much much more clearer to me than it had been before. There might be some uncertainty as to its origin, so I’ll give my take on it. I heard it from Eric Larson a couple of years ago. We had discussed the fact that it should be some easy-to-understand fact about maps of free modules over Noetherian rings. (Many people have known this — Max Lieblich and I had discussed it before; it is alluded to in Nitsure’s excellent presentation in FGA Explained; and I am certain similar discussions have happened many times in the past.)

Eric went home and figured it out, and the next day sent me a short and sweet argument in a pdf file (see below). He had thought he had heard it before, but the only plausible sources he could have heard it from (Joe Harris, or the argument in Eisenbud and Harris) don’t have the argument. But he had certainly heard an argument for Grauert’s Theorem (where the base is reduced). So until I learn otherwise, I suspect he just thought he had heard a full argument of Cohomology and Base Change, and then tried to reconstruct how it should go, and just figured it out. In other words, for now, I believe the argument is original to him. But I would have expected that such a simple argument (especially with its central insight) would have been independently discovered earlier, perhaps many times. However, I am not aware of it in the literature anywhere. Can anyone tell me where it has appeared before? It is such a simple argument that, if I had not been consciously looking for it earlier, I would have believed that I had surely known it myself. So if someone says “I knew that, I just never wrote it down”, I’m not going to value that too highly. On the other hand, there are a number of great explanations of things in algebraic geometry that Dennis Gaitsgory told undergraduates when he was at Harvard; and Nitsure’s argument is presumably written down somewhere; so I know there are things I haven’t seen.

(Incidentally, Ogus told me a fantastically elegant proof of the local criterion for flatness, which was in his first published paper I think, that I love.)

Of course, I’ve digested Eric’s argument into The Rising Sea (and I really wish I hadn’t moved around part of a chapter a couple of years ago, leaving the manuscript in some disarray; I hope to post the new version before too too long). And I’ll eventually post my take on Ogus’ argument here, and I’m also digesting it into the Rising Sea.

One of Eric’s insights, for me, is this. When we generalize the notion of “finite-dimensional vector space” to families, we get finite-rank vector bundles. These have constant rank, but for a coherent sheaf to be a (finite rank) vector bundle, we need more than it be constant rank. Eric suggests that to generalize the notion of “map of finite-dimemnsional vector spaces” in a particularly good way, we want more than a map of vector bundles (as coherent sheaves) — and we want more than that it is of “constant rank”. It is this stronger condition about maps of vector bundles that gives this strong condition of cohomology commuting with base change, which can be applied in different settings (including Grauert, and the Cohomology and Base Change Theorem).

]]>I find this explanation surprisingly comprehensible, and this is really the first time I feel like I understand *why* spectral sequences work, and why they are not mysterious or black magic.

I haven’t been sure what to do with this pdf document, so I’ve just been giving it to people as a “gift”. So consider this to a gift to you too.

It is also posted here. (I am a fan of 3blue1brown.)

]]>But what is moving me to write this today is an email from Zihong Chen, asking about the flawed proof of the Kunneth formula in an earlier version of the notes (since removed, but it may have been after my last posting). I’d scribbled down notes on what I should have said, so I thought I should type it properly, and post it here. (I’m not sure if I will add it to the notes. The price is two pages, which is pretty steep at this point. But it fits squarely into the narrative of the Rising Sea.)

]]>Jarod Alper will be teaching a course on “Introduction to stacks and moduli” at the University of Washington, and the course website is here: https://sites.math.washington.edu/~jarod/math582C.html

I’m going to attend, and I think this course could be as influential as Martin Olsson’s course on stacks back in the Beforetimes (which was central to his development of his wonderful book). Jarod is allowing me to advertise it here. He told me: “Right now I’ve posted a very lengthy introduction & motivation, which I’ll spend all of about one lecture on. I will be gradually posting (and revising) the notes during the quarter.”

The class meets Mon/Wed 11:30-12:50 (beginning Mon Jan 4, 2021), Pacific time. On the course website, he writes: “If you would like to participate informally in the class, please send me an email at jarod@uw.edu with (1) your name, (2) email address, (3) affiliation (if any), (4) status (e.g. 3rd yr PhD student, postdoc, …), and (5) a one sentence summary of your background in algebraic geometry.”

For Stanford folks: I am strongly recommending that most of my current students attend this (with some exceptions depending on their interests), and also recommending that many of my “maybe-they-are-my-students” consider taking it.

]]>Nikolas Kuhn gave a slick answer to this question. Consider the cuspidal curve in the plane (where is of course a field), which has normalization . Then if you remove , you get an affine open subset (why?), but that set is not even set-theoretically cut out by a single equation (hint: pull the equation back to be a function of , but notice that this equation doesn’t lie in the subring .

Entertaining follow-up question: this affine open set is for some ring . What is ?

]]>But I’m inclined to switch over completely to this phrase. The clearest downside is that the notation refers to “Distinguished”. But I already prefer to think of it as the “Doesn’t-vanish” set.

This might suggest a better name for the following two “topologies”: the topology on consisting of principal/distinguished open subsets; and the “topology” on a scheme where the allowed open subsets are affine open subsets, and the allowed open morphisms are “principal/distinguished” inclusions. I’ve been calling the latter the “Distinguished Affine Topology”, and I’m not sure if I gave a name to the former. Are there better names for these?

Separately, now that AGITTOC (at least the first incarnation) is over, and I might write more here, as a world-readable (and world-commentable) notebook.

I’ve not posted a new version of the notes in a long time, because parts of it have been “closed for construction”. But I might resume soon. It might be easiest to post just a few chapters from the beginning, and gradually move forward.

As always, there are many comments here that I’ve not responded to. You might be surprised to find that I’ve actually read them, and made changes in response to many, and have intended changes in response to others.

]]>I was afraid this would happen — I forgot to set my alarms last night, and just woke up, and won’t have things sufficiently ready for today’s scheduled pseudolecture. So I’ll have to postpone it.

There are still two more to go. The next one will *not* be next week — it will be the week after next (Saturday October 3). There is a new seminar, approximately monthly, by Dawei Chen and Qile Chen at Boston College, with two talks , and the second will conflict with AGGITOC’s regular time: https://sites.google.com/bc.edu/map/home . There will be some people interested in both events. (Next week’s second speaker is Hannah Larson, who is definitely worth catching, incidentally.)

I’ve posted this on zulip and the Algebraic Geometry Discord, so I hope this reaches everyone!

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